Predicting the Structural Stability and Formability of
ABO3-Type Perovskite Compounds Using Artificial
Neural Networks
Y M Zhanga, R Ubicb, D F Xuec and S Yangd*
a
Ningbo Institute of Material Technology & Engineering,
Chinese Academy of Sciences, Ningbo, Zhejiang Province, P.R. China.
b
Department of Materials Science & Engineering, Boise State University
1910 University Drive, Boise, Idaho 83725, USA
c
State Key Laboratory of Rare Earth Resource Utilization, Changchun Institute
of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022,
China
d
Engineering Sciences, Faculty of Engineering and the Environments,
University of Southampton, Southampton, SO17 1BJ, UK.
* Corresponding author. Email: s.yang@soton.ac.uk
Abstract
Information on crystal structure is a prerequisite for materials modelling. Several methods
have been developed in the last few years to determine crystal structures of perovskites, an
important structure with diverse physical/chemistry properties. In this work, an artificial
neural networks (ANNs) modelling method is used to make predictions of global instability
index (GII) from bond-valence based tolerance factors tBV. It is found that, by normalizing
with the valence of A-site cations, it is possible to deduce a unified correlation between GII
and tBV for general ABO3-type perovskites. The ANNs were also used to make predictions
about the formability of perovskites by using A-O and B-O bond distances. This method
improves prediction accuracy. In the future it is hoped that the introduction of more
parameters will increase the prediction accuracy, as ANN has a unique ability to make
predictions in high dimensional spaces.
Key words: crystal structure, perovskite, data mining, artificial neural networks, global
instability index, tolerance factors
1
1.0
Introduction
1.1
Crystal structure determination
Crystal-structure and materials properties are intimately linked together, so obtaining
information on crystal structures continues to be an area of ongoing research. Presently,
crystallographic information is a prerequisite for any extensive materials modelling, making
knowledge of crystal structures a more practically-pressing problem (Morgan and Ceder,
2005).
There are three main different methodologies for determination of crystal structures. (1)
Crystal structures can be determined experimentally. This kind of work began in 1911 by von
Laue and his colleagues (Freidrich et al., 1912); and then Bragg (1913) performed crucial
experiments that determined the structures of some very simple crystals. However, the
determination of such information is very time- and cost- expensive. (2) Nowadays, the
evolution of high-speed computers and the derivation of one-electron potentials, which
greatly simplify many-body interactions, make it possible to predict structures using quantum
mechanic principles, as has been demonstrated by Wolverton et al. (2001), Asta et al. (2001),
Blum and Zunger (2004) and Curtarolo et al. (2005). These modelling efforts have made
computational materials design a real possibility for many years; however, difficulties still
exist: i) As Chelikowsky (2004) wrote in a review: “Although the interactions
in …compounds are well understood, it is not an easy task to evaluate the total energy of
solids…As the energy of an isolated atom is in the order of about 106 eV, but the cohesive
energy only in the order of about 1-10 eV/atom, one must have a method that is accurate to
one part on 106, or better.” ii) The number of particles involved complicates the situation of
evaluating the cohesive energy and makes the computations less accurate than experiment
(Villars, 2000). (3) With the help databases of known structures or models which have
physical meanings, certain regularities, such as laws, rules, principles, factors, tendencies or
patterns, might be found to help predict unknown structures. This method, known as structure
mapping, is one kind of the most successful method for crystal structure prediction and has
2
been review by Burdett and Rodgers (1994), Villars (2000) and Pettifor (2000). Examples of
structure mapping include Mooser-Pearson Maps (1959), Zunger Maps (1980), and Villars
maps (1983, 1984a, b), but the best known are Pettifor maps (1984, 1986). This method
orders the very large database into domains of different structure types and can be used as an
initial guide in the search for new compounds with a required structure type. Combinatorial
methods from both computational (Yu et al. 2009, Hautier et al. 2012) and experimental
(Pullar 2007a, 2007b, Zhang 2007, Rossiny 2008, Scott 2008, Chen 2011a, 2011b)
approach have been used to search new materials and explore crystal-structure and materials
properties correlations. In those situations when a large amount of new materials data is
involved, a simple and fast evaluation method is needed. In this paper the application of ANN
on the parameters of tolerance factor tBV have shown a faster and accurate evaluation of the
structural stability of perovskites.
1.2
Data mining method in crystal structure prediction
The data mining (DM) method has become a powerful tool in different branches of materials
science. This technique offers the possibility of making descriptive and even quantitative
predictions in many areas where traditional approaches have had limited success. Generally,
people try to make predictions through constitutive relations, that is, derived mathematically
from basic laws of physics; however, in many cases, the problems are so complex that
constitutive relations either cannot be derived or are too approximate or intractable for
practical quantitative use. The data mining method is based on the assumption that the useful
constitutive relations exist, and these relations could be derived primarily from data rather
than from basic laws of physics (Morgan and Cerbrand, 2005).
Due to the fact that there is no generally reliable and tractable method to predict structure, and
considering that a lot of structural data exist in crystallographic databases such as ICSD
(Bergerhoff et al., 1983), Linus Pauling files (Villars et al., 1998), CRYSTMET (White et
al., 2002), and others mentioned in the review paper by Allen (1998), crystal structure
3
prediction is an area well suited for data mining. Examples include works by Woodley et al.
(1999), Curtarolo et al. (2003), Fischer et al. (2006), Kazius et al. (2006) Hautier et al. (2011)
and, in particular, Morgan et al. (2003), who developed a clustering algorithm to automate the
construction and testing of Pettifor maps based on data from a materials crystal structure
database.
1.3
Structural stability and formability of ABO3-type perovskite compounds
Due to the diverse physical/chemistry properties of perovskite and perovskite-related
materials, which can be applied in a variety of fields, it is useful to discover the regularities
that govern the formation of perovskite-type compounds, and then use them to further guide
the exploration of new materials with unknown structures (Zhang et al., 2007).
The study of ABO3 compounds has a long history. Megaw (1946) accurately determined the
structure of a number of doubled perovskites by examining high-angle lines on X-ray powder
photographs. Salinas-Sanchez et al. (1992) introduced a parameter called the global instability
index (GII), which can be used to determine the overall structural stability of perovskites.
Giaquinta and Loye (1994) later predicted the perovskite structure for a number of
compounds based on the combination of ionic radii and bond ionicities, predicting the
structure of InMO3 (M = Mn, Fe) with this method. Reaney and Ubic (1999) reviewed the
relationship between the tolerance factor (t) and the temperature coefficient of permittivity (τε)
(and therefore resonant frequency, τf) in perovskites and also discussed the effect on
of
changing t on τε in the perovskite-related solid solution series Ba6-3xNd8+2xTi18O54. Lufaso and
Woodward (2001) used a bond-valence model to calculate ideal A – X and B – X bond
distances in order to calculate the bond-valence based tolerance factor (tBV) which they
proposed as a new criterion of the structural stability of ABO3-type perovskite compounds.
Ye et al. (2002) applied a pattern recognition method and found some regularities in the
formation and the lattice distortion of perovskites. Li et al. (2004) used rA – rB structure maps
4
to study the perovskite formability of 197 ABO3 compounds; and then, with the knowledge of
the importance of the octahedral factor (rB/rO) and tolerance factor (
rA  rO
2 (rB  rO )
), another
structure map was constructed to predict the perovskite formability. Jiang et al. (2006)
investigated the regularities governing lattice constant of ABX3-type compounds by a
statistical regression method and found a good correlation among lattice constant, the
estimated bond length between ion B and ion X (rB + rX) and the ionic-radii tolerance factor
tIR; however, they used ionic radii of all ions A, B and X appropriate for sixfold coordination,
which are inappropriate for A and X. Ubic (2007) later used a different approach based solely
on ionic size and derived a simpler and more accurate empirical relationship between ionic
size and lattice constant. In 2007, Zhang et al., based on the bond-valence model (BVM) and
structure-map technology, investigated the structural stability and formability of ABO3-type
perovskite compounds, established new criteria for the structural stability of such compounds,
and created new structure maps that can be used for exploring novel perovskite-related
compounds. In that work, the calculated global instability indices GIIs are compared with
bond-valence based tolerance factors tBVs for 232 ABO3-type perovskite compounds, and the
results are shown in figure 1. From this result, they found the GII values are close to 0 when
tBV values approach 1, and all GII values are less than 1.2 v.u. for ABO3-type perovskite
compounds.
Also, it was found that the structural stability of ABO3-type perovskite
compounds follows the order A1+B5+O3-type > A2+B4+O3-type > A3+B3+O3-type, which agrees
with the experimental measurements. In 2008, Feng et al. used a method similar to that of Li
et al. (2004) to predict the formability of ABO3 cubic perovskite using the octahedral factor
and tolerance factor. Recently, Verma et al. (2008) developed an equation to predict lattice
constant values for cubic perovskites by using the product of ionic charges and average radii
of ions A, B and X, but the results are much less accurate than those of either Lufaso and
Woodward (2001) or Ubic (2007).
5
This work is based on the structural stability and formability data of ABO3-type perovskites
used by Zhang et al. (2007) and makes use of the neural network data-mining method to 1)
test whether it is possible to use a backpropagation neural network to make the prediction of
GII from tBV within each ABO3-type perovskite subclass (i.e., in A1+B5+O3 type, A2+B4+O3
type, and A3+B3+O3 type), and within the whole ABO3-type perovskite; 2) test whether using a
probabilistic neural network could improve the prediction accuracy for perovskite formability
based on the ideal A-O and B-O bond distances (dA-O and dB-O) compared to the work done by
Zhang et al. (2007).
2.0
Experimental Details
2.1
Configuration of the neural networks
In this work, two different neural networks were used. One is backpropagation neural network
(BPANN), which is versatile and can be used to model mapping problems; the other is
probabilistic neural network (PNN), which is a type of radical basis network suitable for
classification problems.
2.2
BPANN
The construction of the BPANN involves the choices of i) number of neural network layers, ii)
number of neurons in the first hidden layer, iii) method for improving generalization, iv)
partitioning of the database and v) data normalization method. The details of this technique
can be found in previous work (Zhang et al., 2008, 2010a, 2010b, 2011).
2.3
Partitioning of the database for PNN
As for BPANN, the generalizing ability of PNN depends on the training database size. It is
necessary to generalize for the unseen data, so the data set used for training should be large
enough to cover the possible known variation in the problem domain. The parent database
here is partitioned into two sub-sets: training and testing. Following the method used in
6
construction of BPANN, as discussed previously (Zhang et al., 2008), the sets are picked as
equally spaced points throughout the original data. The whole data set is partitioned into five
groups, four groups being used for training, and one group is used for testing. Because the
problem domain is not clear in this work, a loop program was used to redistribute the database
in order to make the training set cover the problem domain. One way that can be used to view
the results is a confusion matrix, where matrix element mij gives the number of times a
sample belonging in class Ci was assigned to Cj. As the result, the distribution was selected on
the basis of the total number of samples incorrectly placed into the class
positives). The ideal value of
2.3
m
ij
m
ij
(false
should be zero.
Choice of spread for PNN
The spread of radial basis functions in PNN need to be chosen. As mentioned in MATLAB
(Mathworks), if spread is near zero, the network acts as a nearest neighbour classifier; as
spread becomes larger, the designed network takes into account several nearby design vectors.
In this work, a looped program is used in order to find the best combination of database
distribution and value of spread.
2.4
Determination of input and output parameters
This work was composed of two different parts: 1) prediction of the global instability indices
(GII) from bond-valence based tolerance factor (tBV) using backpropagation neural network
(BPANN); 2) prediction of the perovskite formability based on the ideal A-O and B-O bond
distances (dA-O and dB-O) using probabilistic neural network (PNN).
In part 1, the bond-valence based tolerance factor (tBV) and global instability indices (GII) are
input and output of BPANN, respectively; in part 2, the inputs are A-O (dA-O) and B-O (dB-O)
bond distances, and the output are the formability of perovskites.
7
3.0
Results
3.1
Prediction of GII from bond tBV
From the three curves in figure 1, it is interesting to see whether there is a correlation between
GII and tBV, that is, can one of them, say tBV, be used to predict the other (GII in this case)?
The results of using tBV to predict GII for A1+B5+O3 type perovskites, A2+B4+O3 type
perovskites and A3+B3+O3 type perovskites are shown in figures 2(a) and 2(c) in which the
blue circles represent the training datasets and the purple stars represent the testing datasets.
The best linear fit equations and regression coefficients, R, show that they give sensible
correlations; R=0.999 for figure 2(a), R=0.997 for figure 2(b) and R=0.999 for figure 2(c).
These three results indicate that correlations exist between the tolerance factor (tBV) and
global instability index (GII) for each subclass of perovskites (i.e., A1+B5+O3, A2+B4+O3,
A3+B3+O3). The reasons will be discussed later.
It is also interesting to know whether a uniform correlation between the tolerance factor (tBV)
and global instability index (GII) exists for the whole class of ABO3 perovskites. In order to
address this question, a similar procedure is used to make predictions of GII from tBV. The
result is shown in figure 2(d). By comparing the best linear fit equation and regression
coefficient, R, in this figure (R=0.949) with the results shown in figure 2(a-c), it is found that
a correlation between GII and tBV for all ABO3 perovskites exists, but is not as
straightforward as those shown for the individual classes of previous separately.
There are three different underlying links between variables that can explain the observed
correlations, as shown in figure 3. The dashed line represents an observed association
between the variables x and y. Some association can be explained by a direct cause-and-effect
link between the variables. In figure 3(a), an arrow running from x to y shows “x causes y”.
When thinking about an association between two variables, lurking variables need to be
considered. figure 3(b) illustrates the common response, in which the observed association
between the variables x and y can be explained by a lurking variable z; and both x and y
8
change in response to changes in z. The common response creates an association even though
there may be no direct causal link between x and y. figure 3(c) shows confounding, in which
both the explanatory variable x and the lurking variable z may influence the response variable
y. Since the effect of x is confounded with the effect of z, the influence of x cannot be
distinguished from the influence of z; also, it cannot be said how strong the direct effect of x
on y is. It is difficult even to tell if x influences y at all.
From bond-valence model, the bond valence of the ith ion with respect to the jth ion, sij, is
calculated using equation (1)
 Rij  d ij 

sij  exp 
 B 
(1)
where dij is the cation-anion bond distance; Rij is empirically determined for each cation-anion
pair based on a large number of well-determined bond distances for the cation-anion pair in
question, and has been tabulated; and B is an empirically determined universal constant with a
value of 0.37.
From the calculated bond valence sij, the atomic valences of ith ion Vi(calc.) can be calculated
from equation (2) by summing the individual bond valences (sij) about each ion:
Vi ( calc)   sij
(2)
j
Then, the discrepancy factor di can be calculated from equation (3)
d i  Vi (OX )  Vi ( calc)
(3)
where Vi(OX) is the formal valence for the ith ion.
The overall structural stability, referred as the global instability index (GII), is determined by
comparing the calculated atomic bond-valence Vi(calc.) and formal valence Vi(OX), and
9
calculated from equation (4)
1/ 2
 N
 
GII   d i2  / N 
 
 i 1
 
(4)
where N is the number of atoms in the asymmetric unit. The bond-valence based tolerance
factor, tBV, shown in equation (5), is calculated like the tolerance factor (
rA  rO
2 (rB  rO )
), but
substituting rA+rO and rB+rO with bond distances A-O (dAO) and B-O (dBO) calculated from the
bond-valence model.
t BV 
d AO
2d BO
(5)
From the analysis above, it can be found that both the global instability index (GII) and bondvalence tolerance factor (tBV) are calculated fundamentally from dij and other parameters, and
so there is a common response, as indicated in figure 3(b), between GII and tBV. This can
explain the results for each ABO3 type (A1+B5+O3, A2+B4+O3, A3+B3+O3) shown in figure 2(a)2(c); however, for the whole range of ABO3-type perovskites, the correlation is a bit poor,
which indicates that there is no unified relationship between tBV and GII for the all types of
ABO3 perovskite; however, if each GII value in figure 1 were normalized by the valence of
A-site cations, a unified correlation would emerge. The result is shown in figure 4. Again,
ANN was used to make the prediction of GII values from tBV and the valences of A-site
cations, and the result is shown in figure 5.
3.2
Prediction of perovskite formability
In this part, a probabilistic neural network (PNN) is used to simulate the work done by Zhang
et al. (2007) to predict the likelihood zone of perovskite compounds being formed, based on
the same parameters they used, i.e., ideal A-O and B-O bond distances derived from bond-
10
valence model (BVM). The predictions are made for each type of perovskite (A1+B5+O3,
A2+B4+O3 and A3+B3+O3) and also for all ABO3-type perovskites collectively. The results are
shown in figure 6(a-d).
Figure 6(a) shows that six A1+B5+O3 structures are located in the wrong zone (six nonperovskites locate in perovskite zone), whereas in Zhang et al.’s (2007) work, seven such
structures are located in the wrong zone. Similarly, figure 6(b) shows 11 incorrect A2+B4+O3
perovskite assignments, whereas Zhang et al.’s (2007) have 17 incorrect predictions; and
figure 6(c) shows four incorrect A3+B3+O3 perovskite assignments where Zhang et al. (2007)
shows two.
From these three comparisons, it is found that in the first two cases, the neural network
performs a bit better in terms of predictive accuracy; however, in Zhang et al.’s work (2007),
the boundaries for separating perovskite and non-perovskite zones are regular, which helped
them to point out the conditions that determine the formability of ABO3-type perovskite
compounds. In the third case, Zhang et al. (2007) made all the predictions of non-perovskite
structures correct but made two incorrect predictions of perovskites, while the neural network
makes predictions of all perovskites correctly, but makes four false predictions of nonperovskites. Overall, the neural network performs worse in this case.
5.0
Conclusions and Further Work
In this work, 1) A neural network modelling method is used to make predictions of global
instability index GII from bond-valence based tolerance factors tBV for 232 ABO3-type
perovskite compounds that Zhang et al. (2007) used and found that correlations exist between
these two parameters within each subclass of ABO3-type perovskite compounds; however, for
the general ABO3-type perovskite, this kind of correlation is not obeyed. It was found that the
valence of A-site cations can be treated as another parameter to induce a unified correlation. 2)
Neural networks have also been used to make the prediction of formability of perovskites by
11
using A-O and B-O bond distances in order to find the possibility of improving prediction
accuracy. It was found that, in terms of the prediction accuracy, neural networks can yield
better performances, but it is difficult to give simple and physically meaningful explanations.
Neural networks have the ability to make predictions in high dimension spaces, i.e., they can
have more than two inputs to make the prediction. For the prediction of formability of ABO 3type perovskite compounds, as mentioned by Zhang et al. (2007), besides the A-O and B-O
bond distances, other factors also contribute to the formation of ABO3-type perovskite
compounds, such as i) bond-valence based tolerance factor tBV, ii) temperature and iii)
pressure; therefore, these parameters can be incorporated into future work in order to improve
the accuracy of predictions, which can be used for practical applications.
12
Captions of Figures
Figure 1
Global instability indices (GII) versus bond-valence based tolerance
factors (tBV) for ABO3-type perovskite compounds. (Redrawn from
Zhang et al., 2007)
Figure 2
Prediction of global instability indices (GII) from bond-valence based
tolerance factors (tBV) for (a) A1+B5+O3; (b) A2+B4+O3; (c) A3+B3+O3 and (d) all
type of ABO3 perovskite compounds.
Figure 3
Explanations for an observed association. The broken lines show an
association. The arrows show a cause-and-effect link. The x and y are
observed variables, and z is a lurking variable (Redrawn from Moore
and McCabe, 1999)
Figure 4
Global instability indices (GII) normalised by the valences of A-site cations
versus bond-valence based tolerance factors (tBV) for ABO3-type perovskite
compounds.
Figure 5
Prediction for perovskite formability of all ABO3-type compounds
from tBV and valences of A ions by ANN.
Figure 6
Prediction for perovskite formability of the (a) A1+B5+O3; (b) A2+B4+O3; (c)
A3+B3+O3 and (d) all type of ABO3 by ANN.
13
1.5
1.4
A¹⁺B⁵⁺O₃
A²⁺B⁴⁺O₃
A³⁺B³⁺O₃
GII (u.v.)
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
0.8
0.85
0.9
1
0.95
t BV
Figure 1
14
1.05
1.1
1.15
A = (0.996) T + (0.00335)
1.2
Predicted from NN (A)
1
Training Data Points
Test Data Points
Best Linear Fit
A=T
0.8
0.6
0.4
0.2
R = 0.999
0
0
0.2
0.4
0.6
0.8
Experimental GII Values (T)
Figure 2(a)
15
1
1.2
A = (0.986) T + (0.00734)
0.9
Training Data Points
Test Data Points
Best Linear Fit
A=T
0.8
Predicted from NN (A)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
R = 0.997
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Experimental GII Values (T)
Figure 2(b)
16
0.7
0.8
0.9
A = (0.998) T + (0.000384)
1.4
1.2
Training Data Points
Test Data Points
Best Linear Fit
A=T
Predicted from NN (A)
1
0.8
0.6
0.4
0.2
R = 0.999
0
0
0.2
0.4
0.6
0.8
1
Experimental GII Values (T)
Figure 2(c)
17
1.2
1.4
A = (0.907) T + (0.0604)
1.4
Training Data Points
Test Data Points
Best Linear Fit
A=T
1.2
Predicted from NN (A)
1
0.8
0.6
0.4
0.2
R = 0.949
0
0
0.2
0.4
0.6
0.8
1
Experimental GII Values (T)
Figure 2(d)
18
1.2
1.4
X
Y
X
Y
Z
Causation
(a)
Common response
(b)
Figure 3
19
Y
X
Z
Confounding
(c)
1.5
1.4
A¹⁺B⁵⁺O₃
A²⁺B⁴⁺O₃
A³⁺B³⁺O₃
GII (u.v.)/Valence of A ion
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
0.8
0.85
0.9
0.95
1
t BV
Figure 4
20
1.05
1.1
1.15
A = (0.998) T + (0.000753)
1.4
1.2
Training Data Points
Test Data Points
Best Linear Fit
A=T
Predicted from NN (A)
1
0.8
0.6
0.4
0.2
R = 0.999
0
0
0.2
0.4
0.6
0.8
1
Experimental GII Values (T)
Figure 5
21
1.2
1.4
Figure 6(a)
Figure 6(b)
23
Figure 6(c)
24
Figure 6(d)
25
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